Universality in quasiperiodic Rayleigh-Bénard convection

Abstract

We study universal scaling properties of quasiperiodic Rayleigh-B\'enard convection in a $^{3}\mathrm{\ensuremath{-}}^{4}$He mixture. The critical line is located in a parameter space of Rayleigh and Prandtl numbers using a transient-Poincar\'e-section technique to identify transitions from nodal periodic points to spiral periodic points within resonance horns. We measure the radial and angular contraction rates and extract the linear-stability eigenvalues (Flouquet multipliers) of the periodic point. At the crossings of the critical line with the lines of fixed golden-mean-tail winding number we determine the universality class of our experimental dynamics using f(\ensuremath{\alpha}) and trajectory-scaling-function analyses. A technique is used to obtain a robust five-scale approximation to the universal trajectory scaling function. Different methods of multifractal analysis are employed and an understanding of statistical and systematic errors in these procedures is developed. The power law of the inflection point of the map, determined for three golden-mean-tail winding numbers, is 2.9\ifmmode\pm\else\textpm\fi{}0.3, corresponding to the universality class of the sine-circle map.